Note that \(j=N(r-1)+n\)  holds and \({\bf {e}}_{j}\in \mathbb {R}^{RN}\) . Similarly, the problem also satisfies the Slater’s condition and the duality gap doesn't exist [1]}. Then, by setting \(\partial \Gamma _2 / \partial {\mathbf { \Theta }}\)  to zero, we have
\(\begin{aligned}{{\mathbf { \theta }}^{\rm opt}} = {\left( {\sum \limits _{r = 1}^R {\sum \limits _{n=1}^N {\chi _{r,n}}{{{\bf {e}}_j}{\bf {e}}_j^H} } + {\bf {\Lambda }}} \right)^{ - 1}}{\mathbf {\nu }}.\end{aligned}\) 
